#### What do you think about this approach for solving a particular type of math problem?

The psychologist Dan Kahneman talks about what he calls System 1 and System 2 thinking. System 1 is intuitive and quick and System 2 is more thoughtful but slower.

One example he gives of System 1 taking over is the following problem. A bat and ball together cost $1.10. If the bat costs a dollar more, what is the price of the ball? Kahneman says that the answer that most people come to immediately is that the ball costs 10 cents. Many will give the answer without even testing it, which would show immediately that it is wrong. Ten cents + $1.10 is $1.20, not $1.10.

The problem is easily solved with elementary algebra, but is there an intuitive non-algebraic approach? Suppose we start with both items being the same price, 55 cents. If we add an amount to the first and subtract the same amount from the second, the total will remain $1.10. Furthermore, the difference between the two prices will be twice the amount added and subtracted. Therefore, if the common amount is 50 cents, we will end up with a difference of $1. So the ball costs 55 cents minus 50 cents equals 5 cents.

This type of calculation comes up a lot when you hear about voting. If one person gets 5% more of the vote than the other person, how much of the vote did they each get. We start with 50% each and add and subtract 5/2% = 2.5% to get 52.5% and 47.5%.

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